disclaimer: To ensure that the notebook can be run from (more or less) any point, I try to load the relevant functions or modules whenever I use them in a cell. This is generally not good practice as it adds unneccesary overhead
In [1]:
%matplotlib inline
%load_ext autoreload
%autoreload
import numpy as np
checkBoard = np.zeros((9,9))
checkBoard[0::2, 1::2] = 1
checkBoard[1::2, 0::2] = 1
print(checkBoard)
Then, we import pyplot and image modules from the ploting library matplotlib. Using it, we can display our checkerboard array in image form:¶
In [2]:
import matplotlib.pyplot as plt
import matplotlib.image as mpimg
plt.imshow(checkBoard, cmap='gray', interpolation='nearest')
plt.show()
As another simple example, we will import the data module image processing library scikit-image and load a small image of a bush.¶
First, we want to print the pixel values:¶
In [3]:
from skimage import data
image_of_a_bush = data.lfw_subset()
image_of_a_bush = image_of_a_bush[0,:,:]
#print the #of dimentions, the shape, and the pixel values of the image
print("The number of dimensions of the image is: ", image_of_a_bush.ndim)
print("The size of the image is: ", image_of_a_bush.shape)
print(image_of_a_bush)
In [4]:
plt.figure(figsize=(1,1))
plt.imshow(image_of_a_bush, cmap='gray', interpolation='nearest')
plt.show()
1. Pixel-level operations¶
Now that we have a sense of what a digital image is, let's start manipulating it. We'll begin with simple pixel-level operations¶
1.1 Basic pixel-level operations¶
Let's look at a more interesting image. From scikit-image data we'll open a example IHC image, and plot it using pyplot.¶
In [4]:
from skimage import data
import matplotlib.pyplot as plt
import numpy as np
image_hist = data.immunohistochemistry()
#check the size of the image
print("The number of dimensions of the image is: ", image_hist.ndim)
print("The size of the image is: ", image_hist.shape)
plt.imshow(image_hist, cmap=plt.cm.gray)
Out[4]:
Seems like we have an RGB image. Let's look at every channel independently.¶
In [5]:
plt.figure(figsize=(15,5))
plt.subplot(131)
plt.gca().set_title('Red channel')
plt.imshow(image_hist[:,:,0], cmap='Reds', interpolation='nearest')
plt.subplot(132)
plt.gca().set_title('Green channel')
plt.imshow(image_hist[:,:,1], cmap='Greens', interpolation='nearest')
plt.subplot(133)
plt.gca().set_title('Blue channel')
plt.imshow(image_hist[:,:,2], cmap='Blues', interpolation='nearest')
plt.show()
In [6]:
#for the moment let's look at only the first color channel
image_hist = image_hist[:,:,0]
plt.gca().set_title('First channel')
plt.imshow(image_hist, cmap=plt.cm.gray)
Out[6]:
We can invert the image using the invert function from scikit-images utilities module:¶
In [7]:
from skimage.util import invert
inverted_image = invert(image_hist)
plt.figure(figsize=(15,5))
plt.subplot(121)
plt.gca().set_title('original image')
plt.imshow(image_hist, cmap=plt.cm.gray)
plt.subplot(122)
plt.gca().set_title('inverted image')
plt.imshow(inverted_image, cmap=plt.cm.gray)
Out[7]:
Let's try some other pixel-level operations. We'll use the Exposure module from scikit image.¶
A gamma correction applies the nonlinear transform $V_{out} = V_{in}^\gamma$.
A log transform applies $V_{out} = log(V_{in}+1)$.
A sigmoid transform applies $V_{out} = \frac{1}{1+e^{gain\cdot(\text{cutoff}-V_{in})}}$.
Equalization transforms the intensity histogram of an image to a uniform distribution. It often enhances the contrast of the image
CLAHE works similarly to equalization thats applied separately to different regions of the image.
Try to apply these by calling the relevant function from skimage.exposure, or by direct calculation.
Play with the different parameters and see how they change the output.
In [8]:
from skimage import exposure
# apply gamma scaling with gamma=2
gamma=2
gamma_corrected = exposure.adjust_gamma(image_hist, gamma)
# apply logarithmic scaling
logarithmic_corrected = exposure.adjust_log(image_hist)
# apply sigmoidal scaling with cutoff=0.4
cutoff = 0.4
sigmoid_corrected = exposure.adjust_sigmoid(image_hist, cutoff=cutoff)
# equalize
equalize_corrected = exposure.equalize_hist(image_hist)
# apply Contrast Limited Adaptive Histogram Equalization (CLAHE)
CLHA_corrected = exposure.equalize_adapthist(image_hist)
plt.figure(figsize=(15,10))
plt.subplot(231)
plt.gca().set_title('original')
plt.imshow(image_hist, cmap=plt.cm.gray)
plt.subplot(232)
plt.gca().set_title('gamma corrected')
plt.imshow(gamma_corrected, cmap=plt.cm.gray)
plt.subplot(233)
plt.gca().set_title('log corrected')
plt.imshow(logarithmic_corrected, cmap=plt.cm.gray)
plt.subplot(234)
plt.gca().set_title('sigmoid')
plt.imshow(sigmoid_corrected, cmap=plt.cm.gray)
plt.subplot(235)
plt.gca().set_title('equalized')
plt.imshow(equalize_corrected, cmap=plt.cm.gray)
plt.subplot(236)
plt.gca().set_title('CLHA corrected')
plt.imshow(CLHA_corrected, cmap=plt.cm.gray)
Out[8]:
1.2 Image filtering¶
Spatial filtering is an image processing technique for changing the intensities of a pixel according to the intensities of some neighborhood of pixels.¶
The Kernel of the filter defines the neighborhood and the weights asigned to each pixel in the neighborhood:¶
This procedure is formally a convolution and is marked by an asterisk: $I_o = I_i\ast f$.
side note: since a convolution in the spatial domain is equivalent to multiplication in the frequency domain. Sometimes it is more computationally reasonable to calculate these in fourier space.
side side note: filtering can also be performed in the frequency domain by directly removing a set of frequencies from an image.
The kernel can be of any shape/size, it is applied to each pixel in the image, and the output is a new, filtered, image. The output image is often called the response to the given filter.¶
Example, local average: 
Filtering is an incredibly versatile tool with which you can emphasize certain features or remove other features.¶
Image processing operations implemented with filtering include smoothing, sharpening, and edge enhancement.¶
To implement different image filters, we will use the filters module from scikit-image¶
1.2.1 Smoothing¶
Smoothing, aka low-pass filtering, is used for removing high-frequency noise from images. Most commonly, a gaussian kernel is used, but others (e.g. local mean/median) work too. We'll see the effect of gaussian filtering.¶
Try to change the value of sigma (width of the gaussian) to see how the output changes.
In [9]:
import matplotlib.pyplot as plt
import numpy as np
from skimage import filters
image_hist = data.immunohistochemistry()
sigma = 2
gauss_filtered_img = filters.gaussian(image_hist, sigma=sigma, multichannel=True)
plt.figure(figsize=(15,8))
plt.subplot(121)
plt.gca().set_title('original image')
plt.imshow(image_hist, cmap=plt.cm.gray)
plt.subplot(122)
plt.gca().set_title('response, gaussian smoothing')
plt.imshow(gauss_filtered_img, cmap=plt.cm.gray)
Out[9]:
1.2.2 Sharpening¶
sharpening is sometimes used to enhance a blurry (i.e. crappy) image.¶
- Start with input image
- Apply gaussian filter with very narrow kernel
- Subtract filtered image from input image to get only high frequency components
- Amplify (alpha) and add high frequency components to original input image
In [10]:
filter_blurred_f = filters.gaussian(gauss_filtered_img, sigma=0.5, multichannel=False)
alpha = 3
sharpened = gauss_filtered_img + alpha * (gauss_filtered_img - filter_blurred_f)
plt.figure(figsize=(15,8))
plt.subplot(121)
plt.gca().set_title('input - blury image')
plt.imshow(gauss_filtered_img, cmap=plt.cm.gray)
plt.subplot(122)
plt.gca().set_title('sharpened')
plt.imshow(sharpened, cmap=plt.cm.gray)
Out[10]:
1.2.3 Edge enhancement¶
Edge detecting filters work by measuring the local spatial gradient of an image. Common types are the Sobel, Prewitt and Roberts.¶
The filters are usually applied to each direction individually and then the total magnitude of the gradient is calculated.¶
$|\nabla| = \sqrt{\nabla_x^2+\nabla_y^2}$
Sobel:¶
Prewitt:¶
Roberts:¶
In [59]:
from skimage import data
image_hist = data.immunohistochemistry()
image_hist = image_hist[:,:,2]
# sobel magnitude
filtered_img = filters.sobel(image_hist)
# sobel horizontal
filtered_img_h = filters.sobel_h(image_hist)
# sobel vertical
filtered_img_v = filters.sobel_v(image_hist)
plt.figure(figsize=(15,16))
plt.subplot(221)
plt.gca().set_title('input image')
plt.imshow(image_hist, cmap=plt.cm.gray)
plt.subplot(222)
plt.gca().set_title('sobel filter response - magnitude')
plt.imshow(filtered_img, cmap=plt.cm.gray)
plt.subplot(223)
plt.gca().set_title('sobel filter response - horizontal edges')
plt.imshow(np.abs(filtered_img_h), cmap=plt.cm.gray)
plt.subplot(224)
plt.gca().set_title('sobel filter response - vertical edges')
plt.imshow(np.abs(filtered_img_v), cmap=plt.cm.gray)
Out[59]:
Direct application of edge detectors often results in somewhat noisy responses. A way to overcome this is by first smoothing the image with a gaussian filter and then applying the edge filter. The width of the gaussian kernel will determine the size of edges the filter detects.¶
In [60]:
from skimage import data
from skimage import feature
image_hist = data.immunohistochemistry()
image_hist = image_hist[:,:,2]
# sobel magnitude
filtered_img = filters.sobel(image_hist)
DoG_img = filters.sobel(filters.gaussian(image_hist,sigma=2))
plt.figure(figsize=(15,16))
plt.subplot(221)
plt.gca().set_title('input image')
plt.imshow(image_hist, cmap=plt.cm.gray)
plt.subplot(223)
plt.gca().set_title('sobel filter response - magnitude')
plt.imshow(filtered_img, cmap=plt.cm.gray)
plt.subplot(224)
plt.gca().set_title('canny filter response - magnitude')
plt.imshow(DoG_img, cmap=plt.cm.gray)
Out[60]:
DoG is the also the beginning of the canny edge detection algorithm:¶
1) Apply DoG filter 2) Apply Non-maximum suppression 3) Apply hysteresis thresholding
1.2.4 Gaussian derivatives¶
Now let's remember that the application of filters is an associative operation (because convolution is linear!). It's equivalent to apply the gaussian and then the gradient filter to the image and to apply the gradient filter to the gaussian, then apply the result to the image, i.e. $\nabla*(G*I) = (\nabla*G)*I$ where $\nabla$ is the gradient (derivative) filter, $G$ is the gaussian filter, and $I$ is the image.
We can generalize this idea and apply the derivative multiple times, to get a family of interesting filters:
$\nabla^n*G$:
Let's use a second order derivative (aka a Laplacian) and make a ridge detector:¶
We'll look at a retinal photo where vasculature is an interesting feature¶
We'll invert the images so that the vasculature appears as bright lines on a dark background¶
In [106]:
import matplotlib.pyplot as plt
import numpy as np
from skimage import img_as_float
import matplotlib.image as mpimg
from skimage.color import rgb2gray
from skimage.util import invert
#this is how we load an image from the hard drive
image_retina = ((img_as_float(mpimg.imread("../Data/RetinalPhoto.png"))))
image_ridges = invert(rgb2gray(image_retina))
plt.figure(figsize=(15,5))
plt.subplot(121)
plt.gca().set_title('retinal photo')
plt.imshow(image_retina, cmap=plt.cm.gray)
plt.subplot(122)
plt.gca().set_title('inverted grayscale image')
plt.imshow(image_ridges, cmap=plt.cm.gray)
Out[106]:
We find ridges in the inverted image by applying a 2nd order gaussian derivative filter. The second order derivative is called a Laplacian:¶
In [109]:
sigma=3
LoG_ridges = filters.laplace(filters.gaussian(image_ridges, sigma=sigma))
plt.imshow(LoG_ridges,cmap='gray')
plt.gca().set_title('retinal photo - response to LoG')
Out[109]:
This commonly used ridge detector is called a Laplacian of Gaussian (LoG)!
(We'll get back to this image in a bit)
1.3 Masking¶
A mask is a binary image (0s and 1s) that typically separates a given input image into Foreground (interesting) and Background (boring) regions, or for picking a region-of-interest (ROI). A mask is applied to an image by element-wise multiplication. The size of a mask must be identical to the size of the image it's applied to.¶
Let's begin by creating a simple circular mask. We'll create an array where the value at each point is it's distance from the center of the image, and display it as an image:¶
In [78]:
import matplotlib.pyplot as plt
import numpy as np
#dimensions in x and y
y = 512
x = 512
#position of center
centY = np.ceil(y/2)
centX = np.ceil(x/2)
#create the grid
yy,xx = np.indices((y,x))
#create radial distance map
radialDist = np.sqrt((xx-centX)**2 + (yy - centY)**2)
#display
plt.gca().set_title('Radial distance')
plt.imshow(radialDist, cmap='gray', interpolation='nearest')
plt.show()
Of these points, we'll pick a circle of radius 100 and display it as an image:¶
In [79]:
circ1 = radialDist < 100
plt.show()
plt.gca().set_title('Circle with radius 100')
plt.imshow(circ1, cmap='inferno', interpolation='nearest')
Out[79]:
This object is a mask. If you multiply this matrix of 0s and 1s with an image of the same size, only the parts that are ==1 will remain¶
Let's apply this mask to our histology image.¶
In [80]:
from skimage import data
image_hist = data.immunohistochemistry()
image_hist = image_hist[:,:,2]
plt.gca().set_title('Masked first channel')
plt.imshow(image_hist*circ1, cmap=plt.cm.gray)
Out[80]:
What happens if we invert the mask?¶
In [81]:
from skimage.util import invert
inverted_mask = invert(circ1)
plt.figure(figsize=(15,5))
plt.subplot(121)
plt.gca().set_title('inverted mask')
plt.imshow(inverted_mask, cmap=plt.cm.gray)
plt.subplot(122)
plt.gca().set_title('inverted masked image')
plt.imshow(image_hist*inverted_mask, cmap=plt.cm.gray)
Out[81]:
Just for closure, let's see what happens when we look at the full RGB image and try to apply the mask
In [82]:
image = data.immunohistochemistry()
plt.imshow(image*circ1, cmap=plt.cm.gray)
Whoops. Seems like something is wrong. Our problem is that numpy didn't know how to multiply a 512x512x3 with a 512x512 mask. Numpy makes solving this very easy by adding a singleton dimension (look up broadcasting in your spare time).
In [83]:
image = data.immunohistochemistry()
plt.gca().set_title('Masked image')
plt.imshow(image*np.expand_dims(circ1,2), cmap=plt.cm.gray)
Out[83]:
1.4 Thresholding¶
1.4.1 Simple thresholding¶
Thresholding an image is the process of setting an intensity (or intensities) for separating the different components of an image.¶
In simplest case, the foreground and background have very different intensities. In that case thresholding is just clustering pixels by their intensity levels.¶
In [84]:
#this function from skimage converts images of integer types into floats, which are easier to work with.
import matplotlib.pyplot as plt
import numpy as np
from skimage import img_as_float
from skimage import data
# First, let's create a noisy image of blobs
image_blobs = img_as_float(data.binary_blobs(length=512, seed=1))
sigma = 0.22
image_blobs += np.random.normal(loc=0, scale=sigma, size=image_blobs.shape)
print("The number of dimensions of the image is: ", image_blobs.ndim)
print("The size of the image is: ", image_blobs.shape)
plt.imshow(image_blobs, cmap=plt.cm.gray)
Out[84]:
To find the right threshold, let's examine a histogram of pixel intensity values¶
In [85]:
plt.hist(image_blobs.flatten(),bins=250)
plt.show()
Pick an appropriate threshold, by eye, and see if you can remove the background. What happens when you increase or decrease the threshold?
In [86]:
thresh = 0.5
mask = image_blobs>thresh
plt.figure(figsize=(15,5))
plt.subplot(131)
plt.gca().set_title('original')
plt.imshow(image_blobs, interpolation='nearest', cmap=plt.cm.gray)
plt.subplot(132)
plt.gca().set_title('mask')
plt.imshow(mask, interpolation='nearest', cmap=plt.cm.gray)
plt.subplot(133)
plt.gca().set_title('masked image')
plt.imshow(image_blobs*mask, interpolation='nearest', cmap=plt.cm.gray)
Out[86]:
Our mask looks ok, but it has a lot of salt & pepper speckle noise. Why is that?¶
We can try and use what we learned before about filtering to clean up our results. What filter should we use?
In [87]:
from skimage import filters
thresh = 0.5
mask = filters.gaussian(image_blobs, sigma=1)>thresh
plt.figure(figsize=(15,5))
plt.subplot(131)
plt.gca().set_title('original')
plt.imshow(image_blobs, interpolation='nearest', cmap=plt.cm.gray)
plt.subplot(132)
plt.gca().set_title('mask')
plt.imshow(mask, interpolation='nearest', cmap=plt.cm.gray)
plt.subplot(133)
plt.gca().set_title('masked image')
plt.imshow(image_blobs*mask, interpolation='nearest', cmap=plt.cm.gray)
Out[87]:
It's usually a good idea before creating a mask to despeckle an image using a narrow gaussian filter!
1.4.2 Morphological operations¶
Morphology is a broad set of image processing operations that process images based on shapes. In a morphological operation, each pixel in the image is adjusted based on the value of other pixels in its neighborhood. By choosing the size and shape of the neighborhood, you can construct a morphological operation that is sensitive to specific shapes in the input image. (explanation from Mathworks)
Morphological operations are based around a structuring element, which is a small binary image, often of a disk or a square. The structuring element is positioned at all possible locations in the image and it is compared with the corresponding neighbourhood of pixels. Some operations test whether the element "fits" within the neighbourhood, while others test whether it "hits" or intersects the neighbourhood.
Common operations for image processing
Erosion - output image =1 wherever the structuring element fits (erodes the mask)
Dilation - output image =1 wherever the structuring element hits (expands the mask)
Opening - Erosion followed by dilation (opens gaps in spots where the mask is weakly connected)
Closing - Dilation followed by erosion (closes holes in the mask)
A very thorough explanation of morphological operationscould be found here
In [88]:
from skimage.morphology import erosion, dilation, opening, closing
from skimage.morphology import disk
#define a "disk" structuring element
selem = disk(10)
erosion_mask = erosion(mask, selem)
dilation_mask = dilation(mask, selem)
opening_mask = opening(mask, selem)
closing_mask = closing(mask, selem)
plt.figure(figsize=(15,10))
plt.subplot(231)
plt.gca().set_title('mask')
plt.imshow(mask, interpolation='nearest', cmap=plt.cm.gray)
plt.subplot(232)
plt.gca().set_title('erosion')
plt.imshow(erosion_mask, interpolation='nearest', cmap=plt.cm.gray)
plt.subplot(233)
plt.gca().set_title('dilation')
plt.imshow(dilation_mask, interpolation='nearest', cmap=plt.cm.gray)
plt.subplot(235)
plt.gca().set_title('opening')
plt.imshow(opening_mask, interpolation='nearest', cmap=plt.cm.gray)
plt.subplot(236)
plt.gca().set_title('closing')
plt.imshow(closing_mask, interpolation='nearest', cmap=plt.cm.gray)
Out[88]:
In [89]:
import matplotlib.pyplot as plt
import numpy as np
from skimage import img_as_float
import matplotlib.image as mpimg
#this is how we load an image from the hard drive
image_nuclei = img_as_float(mpimg.imread("../Data/xy040-1.png"))
fig = plt.figure(num=None, figsize=(7.1, 4.6), dpi=80, facecolor='w', edgecolor='k')
print("The number of dimensions of the image is: ", image_nuclei.ndim)
print("The size of the image is: ", image_nuclei.shape)
plt.imshow(image_nuclei, cmap=plt.cm.gray, vmin=0, vmax=0.01)
Out[89]:
Again, let's plot a histogram of intensity values¶
In [90]:
plt.hist(image_nuclei.flatten(),bins=250)
plt.xlim((0, 0.02))
plt.show()
And again we'll pick a value by eye:¶
In [91]:
thresh = 0.003
#remember to despeckle before creating a mask!
mask = filters.gaussian(image_nuclei, sigma=1)>thresh
plt.figure(figsize=(8,15))
plt.subplot(311)
plt.gca().set_title('original')
plt.imshow(image_nuclei, interpolation='nearest', cmap=plt.cm.gray, vmin=0, vmax=0.01)
plt.subplot(312)
plt.gca().set_title('mask')
plt.imshow(mask, interpolation='nearest', cmap=plt.cm.gray)
plt.subplot(313)
plt.gca().set_title('masked image')
plt.imshow(image_nuclei*mask, interpolation='nearest', cmap=plt.cm.gray, vmin=0, vmax=0.01)
Out[91]:
Not bad! But also not very scalable. If we have 100s of images we can't look at them one-by-one and find thresholds by eye. Next, we'll look at some methods for automatically finding the thresholds.¶
1.5 Automated threshold calculation¶
There is a very large list of algorithms for threshold calculation out there that are optimized for different situations. We will briefly review a few of the most common ones.¶
1.5.1 Iterative mean thresholding¶
Algorithm:
- Start with some threshold $T_i$
- Compute the means $m_0$ and $m_1$ of the BG and FG
- Update $T_{i+1} = \frac{m_0+m_1}{2}$
- Repeat until it converges
1.5.2 Otsu thresholding¶
The algorithm exhaustively searches for the threshold that minimizes the intra-class variance, defined for a given threshold $T$ as a weighted sum of variances of the two classes: $\sigma^2_w(T)=\omega_0(T)\sigma^2_0(T)+\omega_1(T)\sigma^2_1(T)$
For 2 classes, minimizing the intra-class variance is equivalent to maximizing inter-class variance, which is much easier to calculate: \begin{align} \sigma^2_b(T) & =\sigma^2-\sigma^2_w(T)=\omega_0(\mu_0-\mu_T)^2+\omega_1(\mu_1-\mu_T)^2 \\ & =\omega_0(T) \omega_1(T) \left[\mu_0(T)-\mu_1(T)\right]^2 \end{align}
1.5.3 Triangle thresholding¶
Algorithm:
- Draw a straight line between the histogram peak and the brightest value.
- From every point on that line, draw the shortest connecting line to the histogram.
- Find longest of these connecting lines.
- Threshold is set at the intersection of that line and the curve.
note: Triangle thresholding is good for situations where the image is mostly background, and there is no clear "peak" of bright pixels.
scikit-image's filters module implements a large variety of thresholding algorithms. Let's apply the ones we just learned about.¶
In [101]:
from skimage import filters
meanThresh = filters.threshold_mean(image_nuclei)
print(meanThresh)
OtsuThresh = filters.threshold_otsu(image_nuclei)
print(OtsuThresh)
TriThresh = filters.threshold_triangle(image_nuclei)
print(TriThresh)
Let's look at the resulting masks we get with each of these thresholds¶
In [102]:
fig = plt.figure(num=None, figsize=(12, 8), dpi=80)
ax1 = fig.add_axes([0.1,0.6,0.4,0.4])
ax1.hist(image_nuclei.flatten(),bins=250)
ax1.axvline(meanThresh, color='g', linestyle='--')
ax1.axvline(OtsuThresh, color='r', linestyle='--')
ax1.axvline(TriThresh, color='k', linestyle='--')
ax1.legend(['mean' ,'otsu', 'triangle'])
ax1.set_title('histogram')
ax2 = fig.add_axes([0.6,0.6,0.4,0.4])
mask_mean = filters.gaussian(image_nuclei, sigma=1)>meanThresh
ax2.imshow(mask_mean)
ax2.set_title('Iterative mean')
ax2.set_axis_off()
ax2 = fig.add_axes([0.1,0.1,0.4,0.4])
mask_otsu = filters.gaussian(image_nuclei, sigma=1)>OtsuThresh
ax2.imshow(mask_otsu)
ax2.set_title('Otsu')
ax2.set_axis_off()
ax2 = fig.add_axes([0.6,0.1,0.4,0.4])
mask_tri = filters.gaussian(image_nuclei, sigma=1)>TriThresh
ax2.imshow(mask_tri)
ax2.set_title('Triangle')
ax2.set_axis_off()
Let's briefly look back at the retinal images and try to mask only the vasculature:¶
load the image and apply LoG filter:
In [103]:
import matplotlib.pyplot as plt
import numpy as np
from skimage import img_as_float
import matplotlib.image as mpimg
from skimage.color import rgb2gray
#this is how we load an image from the hard drive
image_ridges = rgb2gray(img_as_float(mpimg.imread("../Data/RetinalPhoto.png")))
plt.imshow(image_ridges, cmap=plt.cm.gray)
plt.gca().set_title('retinal photo')
sigma = 3
LoG_ridges = filters.laplace(filters.gaussian(invert(image_ridges), sigma=sigma))
ax2.set_axis_off()
Now, let's do the same procedure of automatically finding thresholds:
In [112]:
from skimage import filters
meanThresh = filters.threshold_mean(LoG_ridges)
print(meanThresh)
OtsuThresh = filters.threshold_otsu(LoG_ridges)
print(OtsuThresh)
TriThresh = filters.threshold_triangle(LoG_ridges)
print(TriThresh)
fig = plt.figure(num=None, figsize=(12, 8), dpi=80)
ax1 = fig.add_axes([0.1,0.6,0.4,0.4])
ax1.hist(image_nuclei.flatten(),bins=250)
ax1.axvline(meanThresh, color='g', linestyle='--')
ax1.axvline(OtsuThresh, color='r', linestyle='--')
ax1.axvline(TriThresh, color='k', linestyle='--')
ax1.set_title('histogram')
ax1 = fig.add_axes([0.2,0.65,0.3,0.3])
plt.imshow(image_ridges, cmap=plt.cm.gray)
ax1.set_axis_off()
ax1.legend(['mean' ,'otsu', 'triangle'])
ax2 = fig.add_axes([0.6,0.6,0.4,0.4])
mask_mean = LoG_ridges>meanThresh
ax2.imshow(mask_mean)
ax2.set_title('Iterative mean')
ax2.set_axis_off()
ax2 = fig.add_axes([0.1,0.1,0.4,0.4])
mask_otsu = LoG_ridges>OtsuThresh
ax2.imshow(mask_otsu)
ax2.set_title('Otsu')
ax2.set_axis_off()
ax2 = fig.add_axes([0.6,0.1,0.4,0.4])
mask_tri = LoG_ridges>TriThresh
ax2.imshow(mask_tri)
ax2.set_title('Triangle')
ax2.set_axis_off()
Play around with the width of the gaussian. Which thresholding algorithm works best in this case?
1.5.4 Local thresholding¶
All of the methods we saw so far are global in the sense that the same threshold is applied to the whole picture. Sometimes we can have an image with vastly different intensity distributions at different locations. Using local thresholding, we can overcome such cases.¶
Let's compare the results from a global (Otsu) and a local threshold.
In [94]:
from skimage import data
image = data.page()
fig = plt.figure(num=None, figsize=(12, 8), dpi=80)
#global thresholding
threshGlobal = filters.threshold_otsu(image)
ax1 = fig.add_axes([0.1,0.6,0.4,0.4])
ax1.set_title('mask - Otsu threshold')
plt.imshow(image ,cmap='gray')
ax2 = fig.add_axes([0.6,0.6,0.4,0.4])
ax2.set_title('mask - Otsu threshold')
plt.imshow(image>threshGlobal,cmap='gray')
#local thresholding
#Try and change this number and see what happens
block_size = 81
threshLocal = filters.threshold_local(image, block_size)
ax1 = fig.add_axes([0.1,0.2,0.4,0.4])
ax1.imshow(threshLocal,cmap='gray')
ax1.set_title('local threshold map')
ax2 = fig.add_axes([0.6,0.2,0.4,0.4])
ax2.set_title('mask - Local threshold')
plt.imshow(image>threshLocal,cmap='gray')
Out[94]:
2. Image segmentation¶
Image segmentation is the process of partitioning a digital image into multiple segments. The goal of segmentation is to simplify and/or change the representation of an image into something that is more meaningful and easier to analyze.¶
2.1 Connected components¶
After we generate a mask, the simplest segmentation involves simply looking for regions in the mask that are connected and labeling each one as a separate object.¶
We begin by generating a simple mask using the triangle threshold method:¶
In [23]:
import matplotlib.pyplot as plt
import numpy as np
from skimage import img_as_float
import matplotlib.image as mpimg
from skimage import filters
image_nuclei = img_as_float(mpimg.imread("../Data/xy040-1.png"))
TriThresh = filters.threshold_triangle(image_nuclei)
#despeckle
mask = filters.gaussian(image_nuclei, sigma=1)>TriThresh
scikit-image's measure module implements a variety of useful methods for segmentation. the label function returns a labeled image of connected components (CCs). Each CC is uniquely numbered by an integer.¶
$\begin{bmatrix} 1 & 1 & 0 & 0 & 2\\ 1 & 1 & 0 & 2 & 2\\ 0 & 0 & 0 & 0 & 0\\ 0 & 3 & 0 & 4 & 4\\ 0 & 0 & 0 & 4 & 4\\ \end{bmatrix}$
In [24]:
from skimage import measure
labels = measure.label(mask)
plt.figure(figsize=(12,5))
plt.subplot(121)
plt.imshow(mask, cmap='gray')
plt.subplot(122)
plt.imshow(labels, cmap='nipy_spectral')
Out[24]:
We can easily generate a mask for a specific CC using the binary operation labels==i¶
In [25]:
i=43
mask_of_CC_i = labels==i
plt.figure(figsize=(10,5))
plt.imshow(mask_of_CC_i, cmap='gray')
Out[25]:
In [26]:
i=87
mask_of_CC_i = labels==i
plt.imshow(mask_of_CC_i, cmap='gray')
Out[26]:
These problems can be partially resolved using morphological operations, but there's no silver bullet¶
In [27]:
from skimage.morphology import erosion, dilation, opening, closing
from skimage.morphology import disk
#define a "disk" structuring element
selem1 = disk(10)
selem2 = disk(7)
plt.figure(figsize=(15,10))
plt.subplot(121)
plt.gca().set_title('original')
plt.imshow(mask, cmap='nipy_spectral')
plt.subplot(122)
plt.gca().set_title('opening')
plt.imshow(dilation(erosion(mask, selem1),selem2), interpolation='nearest', cmap='nipy_spectral')
Out[27]:
2.2 Watershed based segmentation¶
2.2.1 The watershed algorithm¶
The watershed transformation treats the image it operates upon like a topographic map, with the brightness of each point representing its height, and finds the lines that run along the tops of ridges.
More precisely, the algorithm goes as follows:
- Label local minima (i.e. $S_1$, $S_2$)
- Move to next higher intensity level
- Assign to each point the label of it's closest label set. Points equidistant to multiple sets are labeled as boundaries and intensities set to 0
- Repeat until all points are labeled
Let's start with a very naive application. We will invert the image, and then simply apply the watershed function from the scikit-image morphology module. The function returns a labeled image.
In [28]:
import matplotlib.pyplot as plt
import numpy as np
from skimage import img_as_float
import matplotlib.image as mpimg
from skimage import filters
from skimage.util import invert
from skimage.morphology import watershed
image_nuclei = img_as_float(mpimg.imread("../Data/xy040-1.png"))
#invert image
image_to_watershed = invert(image_nuclei)
#Calculate watershed transform
labels_naive = watershed(image_to_watershed, watershed_line = 1)
plt.figure(figsize=(15,10))
#let's look at all the boundaries
plt.figure(figsize=(12,20))
plt.subplot(211)
plt.gca().set_title('image fed to watershed')
plt.imshow(image_to_watershed, cmap='gray')
plt.subplot(212)
plt.gca().set_title('watershed result')
plt.imshow(filters.sobel(labels_naive), cmap='nipy_spectral')
Out[28]:
So this clearly didn't work. Why? How do we fix it?¶
Noise generates a ton of local minima. Each gets its own basin. This leads to massive oversegmentation.
Watershed segmentation is only a part of a segmentation pipeline. Preprocessing (denoising, smoothing, seeding minima) of the image is CRUCIAL for it to work well.¶
The first thing we'll do is to apply the mask that we found before. This is simply done by adding a mask argument to the watershed function.
In [24]:
import matplotlib.pyplot as plt
import numpy as np
from skimage import img_as_float
import matplotlib.image as mpimg
from skimage import filters
from skimage.util import invert
from skimage.morphology import watershed
image_nuclei = img_as_float(mpimg.imread("../Data/xy040-1.png"))
TriThresh = filters.threshold_triangle(image_nuclei)
mask = filters.gaussian(image_nuclei, sigma=1)>TriThresh
#mask and invert image
image_to_watershed = image_nuclei*mask
image_to_watershed = invert(image_to_watershed)
#Calculate watershed transform
#pass the mask to the watershed function so it avoids segmenting the BG
labels_masked = watershed(image_to_watershed, watershed_line = 1, mask=mask)
#let's look at all the boundaries
plt.figure(figsize=(12,20))
plt.subplot(211)
plt.gca().set_title('image fed to watershed')
plt.imshow(image_to_watershed, cmap='gray')
plt.subplot(212)
plt.gca().set_title('watershed result')
plt.imshow(filters.sobel(labels_masked), cmap='nipy_spectral')
Out[24]:
So we got rid of all the BG regions, but we are still oversegmenting. Why?¶
Let's try to smoothen the image and get rid of the many local minima. How wide should the gaussian kernel be?
In [25]:
import matplotlib.pyplot as plt
import numpy as np
from skimage import img_as_float
import matplotlib.image as mpimg
from skimage import filters
from skimage.util import invert
from skimage.morphology import watershed, thin
image_nuclei = img_as_float(mpimg.imread("../Data/xy040-1.png"))
TriThresh = filters.threshold_triangle(image_nuclei)
mask = filters.gaussian(image_nuclei, sigma=1)>TriThresh
#mask, smooth, and invert the image
sigma_for_smoothing = 4
image_to_watershed = image_nuclei*mask
image_to_watershed = filters.gaussian(image_to_watershed, sigma=sigma_for_smoothing)
image_to_watershed = invert(image_to_watershed)
#Calculate watershed transform
#pass the mask to the watershed function so it avoids segmenting the BG
labels_masked_smooth = watershed(image_to_watershed, watershed_line = 1, mask=mask)
#let's look at all the boundaries
plt.figure(figsize=(12,20))
plt.subplot(211)
plt.gca().set_title('image fed to watershed')
plt.imshow(image_to_watershed, cmap='gray')
plt.subplot(212)
plt.gca().set_title('watershed result')
plt.imshow(filters.sobel(labels_masked_smooth), cmap='nipy_spectral')
Out[25]:
We're starting to get somewhere!! Can we do better?¶
We can do more to help the algorithm by providing local markers (seeds) from which to start the process¶
We will find seeds by calculating local maxima over areas that are larger than 30x30 pixels using the footprint argument for the function peak_local_max¶
In [26]:
import matplotlib.pyplot as plt
import numpy as np
from skimage import img_as_float
import matplotlib.image as mpimg
from skimage import filters
from skimage import measure
from skimage.morphology import disk
from skimage.morphology import erosion, dilation, opening, closing
from skimage.util import invert
from skimage.morphology import watershed
from skimage.feature import peak_local_max
image_nuclei = img_as_float(mpimg.imread("../Data/xy040-1.png"))
mask = filters.gaussian(image_nuclei, sigma=1)>TriThresh
#mask, smooth, and invert the image
sigma_for_smoothing = 3
image_to_watershed = image_nuclei*mask
image_to_watershed = filters.gaussian(image_to_watershed, sigma=sigma_for_smoothing)
image_to_watershed = invert(image_to_watershed)
#find local peaks to use as seeds
MaskedImagePeaks = peak_local_max(invert(image_to_watershed), footprint=np.ones((30, 30)), threshold_abs=None, threshold_rel=None, exclude_border=False, indices=False).astype('int')
#create disk structuring element of radius 5
selem = disk(5)
#dilate local peaks so that close ones merge
peakMask = dilation(MaskedImagePeaks,selem)
# label local peak regions to find initial markers
markers = measure.label(peakMask)
#pass the *markers* argument to the watershed function
labels_localmax_markers = watershed(image_to_watershed,markers, watershed_line = 1, mask=mask)
#let's look at all the boundaries
plt.figure(figsize=(12,20))
plt.subplot(211)
plt.gca().set_title('image fed to watershed')
plt.imshow(image_to_watershed-peakMask, cmap='gray')
plt.clim((0.95, 1))
plt.subplot(212)
plt.gca().set_title('watershed result')
plt.imshow(filters.sobel(labels_localmax_markers), cmap='nipy_spectral')
Out[26]:
This is pretty good! We're still getting a few errors here and there, but there's no big systematic over- or under- segmentation. This is a typical good result when dealing with real data.¶
3. Feature extraction¶
Feature extraction is a process of dimensionality reduction by which an initial raw image is reduced to a list of objects and attributes¶
3.1 Extracting region properties¶
scikit-image's measure module implements a method called regionprops that accepts a labeled mask of connected components, and, optionally, a corresponding image, and returns a list. Each object on the list contains useful data about the size, shape, position, and intensity (see the full list here) of a specific component.¶
The length of the list is equal to the total number of objects detected.
We'll start by extracting the number of CC we found and the area of each CC¶
In [6]:
from skimage import measure
#We use regionprops
props = measure.regionprops(labels_localmax_markers,image_nuclei)
#how many total connected components did we get?
print(len(props))
props[1].perimeter
Out[6]:
In [7]:
areas = [r.area for r in props]
intensities = [r.mean_intensity for r in props]
#let's look at all the boundaries
plt.figure(figsize=(12,6))
plt.subplot(121)
plt.gca().set_title('areas')
plt.hist(areas)
plt.subplot(122)
plt.gca().set_title('intensities')
plt.hist(intensities)
Out[7]:
We can look at indivudual object found
In [9]:
i=10
plt.imshow(props[i].intensity_image)
plt.gca().set_title('Single cell')
Out[9]:
Let's use a scatter plot to compare our results to the image
In [10]:
intensities = np.array([r.mean_intensity for r in props])
centroids = np.array([r.centroid for r in props])
fig = plt.figure(figsize=(12,15))
fig.add_axes([0.1,0.6,0.4,0.25])
plt.gca().set_title('original')
plt.imshow(image_nuclei, interpolation='nearest', cmap=plt.cm.gray, vmin=0, vmax=0.02)
fig.add_axes([0.6,0.6,0.4,0.25])
plt.scatter(centroids[:,1],centroids[:,0], c=intensities)
plt.axis('equal')
plt.gca().invert_yaxis()
Or even nicer scatter plots!
In [11]:
intensities = np.array([r.mean_intensity for r in props])
areas = np.array([r.area for r in props])
centroids = np.array([r.centroid for r in props])
fig = plt.figure(figsize=(12,15))
fig.add_axes([0.1,0.6,0.4,0.25])
plt.gca().set_title('original')
plt.imshow(image_nuclei, interpolation='nearest', cmap=plt.cm.gray, vmin=0, vmax=0.02)
fig.add_axes([0.6,0.6,0.4,0.25])
plt.scatter(centroids[:,1],centroids[:,0], c=intensities, s=areas/20)
plt.axis('equal')
plt.gca().invert_yaxis()
plt.text(centroids[10,1],centroids[10,0],props[10].label)
Out[11]:
You can even draw your points directly on the image!
In [12]:
intensities = np.array([r.mean_intensity for r in props])
areas = np.array([r.area for r in props])
centroids = np.array([r.centroid for r in props])
fig = plt.figure(figsize=(12,15))
fig.add_axes([0.1,0.6,0.8,0.5])
plt.gca().set_title('original')
plt.imshow(image_nuclei, interpolation='nearest', cmap=plt.cm.gray, vmin=0, vmax=0.02)
plt.gca().patch.set_alpha(0.5)
plt.scatter(centroids[:,1],centroids[:,0], c=intensities, alpha=1)
Out[12]:
Let's define some useful functions for converting a list of props into a pandas dataframe. These should become obsolete soon since the new version of scikit-image will have this functionality¶
In [13]:
import pandas as pd
def scalar_attributes_list(im_props):
"""
Makes list of all scalar, non-dunder, non-hidden
attributes of skimage.measure.regionprops object
"""
attributes_list = []
for i, test_attribute in enumerate(dir(im_props[0])):
#Attribute should not start with _ and cannot return an array
#does not yet return tuples
if test_attribute[:1] != '_' and not\
isinstance(getattr(im_props[0], test_attribute), np.ndarray):
attributes_list += [test_attribute]
return attributes_list
def regionprops_to_df(im_props):
"""
Read content of all attributes for every item in a list
output by skimage.measure.regionprops
"""
attributes_list = scalar_attributes_list(im_props)
# Initialise list of lists for parsed data
parsed_data = []
# Put data from im_props into list of lists
for i, _ in enumerate(im_props):
parsed_data += [[]]
for j in range(len(attributes_list)):
parsed_data[i] += [getattr(im_props[i], attributes_list[j])]
# Return as a Pandas DataFrame
return pd.DataFrame(parsed_data, columns=attributes_list)
Now, to get all the properties in table/dataframe form we simply run:¶
In [14]:
props_df = regionprops_to_df(props)
props_df
Out[14]:
Finally, if we imaged our cells in multiple channels, we would want use the same segmented nuclei and measure intensities of other channels.¶
In [40]:
from skimage import measure
from skimage import img_as_float
import matplotlib.image as mpimg
#load another channel
image_2ndChannel = img_as_float(mpimg.imread("../Data/xy040-2.png"))
# extract regionprops using labels_localmax_markers mask from image_2ndChannel
props_other_channel = measure.regionprops(labels_localmax_markers,image_2ndChannel)
plt.figure(figsize=(12,6))
plt.subplot(121)
plt.gca().set_title('Nuclei')
plt.imshow(image_nuclei, cmap='gray')
plt.subplot(122)
plt.gca().set_title('other channel')
plt.imshow(image_2ndChannel, cmap='gray')
Out[40]:
Extract only the intensity related features (others* are features of the mask only)¶
In [ ]:
mean_2nd_channel = [r.mean_intensity for r in props_other_channel]
max_2nd_channel = [r.max_intensity for r in props_other_channel]
min_2nd_channel = [r.min_intensity for r in props_other_channel]
Add these new features to the pandas dataframe¶
In [17]:
props_df['mean_intensity_ch2'] = mean_2nd_channel
props_df['max_intensity_ch2'] = max_2nd_channel
props_df['min_intensity_ch2'] = min_2nd_channel
props_df
Out[17]:
Sometimes it's easier to see a bimodal distribution in log scale
In [56]:
fig = plt.figure(figsize=(12,6))
fig.add_axes([0.1,0.1,0.4,0.4])
plt.gca().set_title('Histogram of intensities')
plt.hist(props_df.mean_intensity_ch2,20)
fig.add_axes([0.6,0.1,0.4,0.4])
plt.gca().set_title('Histogram of log of intensities')
plt.hist(np.log(props_df.mean_intensity_ch2),20)
Out[56]:
We can compare distributions of different channels
In [57]:
plt.figure(figsize=(12,6))
plt.gca().set_title('scatter plot of intensities')
plt.scatter(np.log(props_df['max_intensity_ch2']), np.log(props_df['max_intensity']))
plt.xlabel('Ch2')
plt.ylabel('Ch1')
Out[57]:
